$ \sum_{k=1}^{+\infty} \frac{\sin^2 (kx)}{k}$ and $ \sum_{k=1}^{+\infty} \frac{\cos^2 (kx)}{k}$

Question

Let's examine the series $ \sum\limits_{k=1}^{+\infty} \dfrac{\sin^2 (kx)}{k}$ and $ \sum\limits_{k=1}^{+\infty} \dfrac{\cos^2 (kx)}{k}$

---

My attempt :
$\forall t , ~ \cos^2(t) + \sin^2(t) =1$ and $\sum\limits_{k \ge 1} \dfrac{1}{k} =\infty$ .
As the two terms are positives, at least one of the series should be divergent.
How to prove that both series are divergent ?

As given in hint, $\cos^2(kx)= 1 + 2 \cos(2kx)$

Answer 1

HINT:

Both series diverge. To show this, make use of the identities

$$\begin{align} \sin^2(x)&=\frac{1-\cos(2x)}{2}\\\\ \cos^2(x)&=\frac{1+\cos(2x)}{2} \end{align}$$

along with the fact that $\sum_{n=1}^\infty \frac{\cos(2nx)}{n}$ converges for $x\ne m\pi$, $m\in \mathbb{Z}$, as guaranteed by Dirichlet's test.